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inequalities
Frédéric Bernicot, Dorothee Frey
To cite this version:
Frédéric Bernicot, Dorothee Frey. Riesz transforms through reverse Hölder and Poincaré inequalities. Mathematische Zeitschrift, Springer, 2016, 284 (3), pp.791-826. �hal-01129985�
POINCAR´E INEQUALITIES
FR ´ED ´ERIC BERNICOT, DOROTHEE FREY
Abstract. We study the boundedness of Riesz transforms in Lpfor p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse H¨older inequality, the Riesz transform is always bounded on Lp for p in some interval [2, 2 + ε), and that Lp gradient estimates for the semigroup imply boundedness of the Riesz transform in Lq for q ∈ [2, p). This improves results of [7] and [6], where the stronger assumption of a Poincar´e inequality and the assumption e−tL(1) = 1 were made. The Poincar´e
inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms.
1. Introduction
The Lp boundedness of Riesz transforms on manifolds has been widely studied in
recent years. The Riesz transform has to be thought of as “one side” (or one half) of the commutator between two first order operators: the gradient, coming from the metric structure of the underlying space, and the square root of a second order operator under consideration (e.g. the Laplace-Beltrami operator on a Riemannian manifold). Thus, Riesz transform bounds allow to compare the two corresponding first order homogeneous Sobolev spaces. The aim of this article is to give new sufficient criteria for the boundedness of Riesz transforms in Lp for p > 2, and to
study its stability under elliptic perturbations.
Before describing our results in detail, let us introduce the setting and some notation.
1.1. Setting. Let (X, d) be a locally compact separable metric space, equipped with a Borel measure µ, finite on compact sets and strictly positive on any non-empty open set. For Ω a measurable subset of X, we shall denote µ (Ω) by |Ω|. For all x ∈ X and all r > 0, denote by B(x, r) the open ball for the metric d with centre x and radius r, and by V (x, r) its measure |B(x, r)|. For a ball B of radius r and a real λ > 0, denote by λB the ball concentric with B and with radius λr. We shall sometimes denote by r(B) the radius of a ball B. We will use u . v to say that there exists a constant C (independent of the important parameters) such
Date: March 9, 2015.
2010 Mathematics Subject Classification. 58J35, 42B20.
Key words and phrases. Poincar´e inequalities, Riesz transform, Reverse H¨older inequality, har-monic functions.
This project is partly supported by ANR projects AFoMEN no. 2011-JS01-001-01 and HAB no. ANR-12-BS01-0013.
that u ≤ Cv, and u ' v to say that u . v and v . u. Moreover, for Ω ⊂ X a subset of finite and non-vanishing measure and f ∈ L1loc(X, µ),R−
Ωf dµ = 1
|Ω|R f dµ
denotes the average of f on Ω.
From now on, we assume that (X, d, µ) is a doubling metric measure space, which means that the measure µ satisfies the doubling property, that is
(VD) V (x, 2r) . V (x, r), ∀ x ∈ X, r > 0.
As a consequence, there exists ν > 0 such that
(VDν) V (x, r) .
r s
ν
V (x, s), ∀ r ≥ s > 0, x ∈ X,
We abstractly define a gradient operator Γ, in terms of which we express first order regularity on the metric space.
Assumptions on Γ. Assume that there exists a sublinear operator Γ, with dense
domain F ⊂ L2(X, µ). Assume that Γ is a local operator, which means that for
every f ∈ F ,
supp Γf ⊆ supp f.
Moreover, assume that Γ satisfies a relative Faber-Krahn inequality, that is for every ball B of radius r ≤ δ diam(X) with some δ < 1, and all f ∈ F supported in B, (1.1) Z B |f |2dµ 1/2 . r Z B |Γf |2dµ 1/2 .
We then consider an unbounded operator L on L2(X, µ) under the following
assumptions.
Assumptions on L. Assume that L is an injective, ω-accretive operator with domain D ⊂ F in L2(X, µ), where 0 ≤ ω < π/2. Assume that for all f ∈ D,
(R2) kΓf k2 . kL1/2f k2.
Assume that L satisfies L2 Davies-Gaffney estimates, which means that for every
r > 0 and all balls B1,B2 of radius r
(DG) ke−r2LkL2(B 1)→L2(B2)+ krΓe −r2L kL2(B 1)→L2(B2) . e −cd2(B1,B2) r2 .
By our assumptions, L is an injective, maximal accretive operator on L2(X, µ),
and therefore has a bounded H∞ functional calculus on L2(X, µ). The assumption
ω < π2 implies that −L is the generator of an analytic semigroup in L2(X, µ). See
[1, 33] for definitions and further considerations, and Section 2 for examples. We will assume the above throughout the paper. We abbreviate the setting with (X, µ, Γ, L).
1.2. Notation. For a (sub)linear operator T and two exponents 1 ≤ p ≤ q ≤ ∞, we say that T satisfies Lp-Lq off-diagonal estimates at scale r > 0 if there exist
implicit constants such that for all balls B1,B2 of radius r and every function
f ∈ Lp(X, µ) supported in B 1, we have − Z B2 |T f |qdµ 1/q . exp −cd 2(B 1, B2) r2 − Z B1 |f |pdµ 1/p .
Denote
pL:= infp ∈ (1, 2]; for all t > 0, e−tL satisfies Lp-L2 off-diagonal estimates ,
pL:= supp ∈ [2, ∞); for all t > 0, e−tL satisfies L2-Lp off-diagonal estimates . We assume that pL6= 2 and pL 6= 2.
By composing off-diagonal estimates, it follows that for every p, q ∈ (pL, pL) with
p ≤ q, the semigroup e−tL satisfies Lp-Lq off-diagonal estimates. We deduce that for every p ∈ (pL, pL),
sup
t>0
ke−tLkp→p < ∞,
and (e−tL)t>0is bounded analytic on Lp(X, µ), see [15, Corollary 1.5]. In particular,
it means that (tLe−tL)t>0 is bounded on Lp(X, µ) uniformly in t > 0.
For p ∈ (1, ∞), one says that (Rp) holds if the Riesz transform R := ΓL−1/2 is
bounded on Lp(X, µ), which means
(Rp) kΓf kp . kL1/2f kp, ∀ f ∈ D,
and that estimates (Gp) on the gradient of the semigroup holds if
(Gp) sup
t>0
k√tΓe−tLkp→p < +∞.
We refer the reader to [7] for the introduction of this notion, and where the inves-tigation of the link between (Gp) and (Rp) has started. In the following, whenever
we talk about (Gp), we shall implicitly assume p ∈ (pL, pL).
Let us now formulate scale-invariant Poincar´e inequalities on Lp(X, µ), which may or may not be true. More precisely, for p ∈ [1, +∞), one says that (Pp) holds
if (Pp) − Z B |f − − Z B f dµ|pdµ 1/p . r − Z B |Γf |pdµ 1/p , ∀ f ∈ F ,
where B ranges over balls in X of radius r. Recall that (Pp) is weaker and weaker
as p increases, that is (Pp) implies (Pq) for q > p, see for instance [30]. The version
for p = ∞ is trivial in the Riemannian setting.
1.3. Main results and state of the art. As started in [7], it is natural to study the connection between gradient estimates (Gp) and the boundedness of the Riesz
transform (Rp). It is easy to see that for every p ∈ (pL, pL), (Rp) implies (Gp) by
analyticity of the semigroup in Lp(X, µ). For p = 2, we have (R
2) by assumption,
and therefore also (G2). Following [21, 14, 31], it is known that (Gp) and (Rp) also
hold for every p ∈ (pL, 2).
But the following question still remains open:
Question A. For p ∈ (2, pL), does (G
p) imply (Rp) or at least (Rq) for every
q ∈ (2, p)?
This question is still open in the general framework we are considering here, and no counter-example is known. We refer the reader to the last paragraph of Subsection 2.1 where we describe two situations where (Gp) is known but not (Rp).
So it is a very deep question to understand which extra property (or maybe none) is sufficient / necessary to deduce (Rp) from (Gp).
In [7], a positive answer has been obtained under the additional assumption of the Poincar´e inequality (P2) and the conservation property. Here, we say that L
satisfies the conservation property, if e−tL(1) = 1 for all t > 0 (1), and that Γ
satisfies the conservation property, if for every φ ∈ F , one has Γφ = 0 on every ball B such that φ is constant on B. With our notation, the result in [7] states as Theorem ([7]). Let (X, µ, Γ, L) be as in Section 1.1. Assume (P2), pL = ∞, and
assume that L and Γ satisfy the conservation property. Suppose p ∈ (2, ∞). If (Gp)
holds, then (Rq) holds for every q ∈ (2, p).
However, in order to have (Rp) for some p > 2, it is known that neither (P2)
is a necessary assumption, as the example of [17] for the two copies of Rn glued
smoothly along their unit circles shows, nor is the conservation property, as the example of some Schr¨odinger operator shows.
In the present work, we shall push the argument of [7] further by weakening these two assumptions (P2) and conservation property. We consider two situations, first
the more specific situation of a non-negative, self-adjoint operator, and then, in a second part, the situation of a sectorial operator as described above. In both cases, we are able to relax on the assumptions imposed in [7].
Let us first discuss the case of non-negative, self-adjoint operators L. Here, we can in particular make use of the fact that the assumption (DG) is equivalent to the finite propagation speed property of√L. This fact was observed in [38], and it was used there in order to study the boundedness of the Riesz transform for p < 2, as well as for p > 2 on 1-forms. We then introduce a property (RHp) (see Definition
2.8), which describes a reverse local H¨older inequality for the gradient of harmonic functions. This property already appeared in [27] for the solutions of elliptic PDEs. In the context of Riesz transform bounds, it was already used in various results for Schr¨odinger operators [40, 5, 39], and in [6] for the Laplace-Beltrami operator on a Riemannian manifold. In Subsection 3.4, we prove the following.
Theorem 1.1. Let (X, µ, Γ, L) be as in Section 1.1. Assume that L is a non-negative, self-adjoint operator, with pL> ν or with the additional assumption (2.6).
Suppose p ∈ (2, pL). If (G
p) and (RHp) hold, then (Rq) holds for every q ∈ (2, p).
Following [12, Theorem 6.3], we know that if L and Γ satisfy the conservation property, the combination (Gp) with (Pp) for some p > 2 implies (P2), and so
implies (Rq) for every q ∈ (2, p) by [7].
Since we will also check that (Gp) with (Pp) (and in particular (P2)) implies
(RHp) (see Proposition 3.9), this new result improves [7]. Moreover, we remove the
assumption of the conservation property. This allows to apply the result to e.g. Schr¨odinger operators. However, Question A still remains open in its generality. since for the two glued copies of Rn with its Riemannian gradient and the Laplace
operator, we have (Gp) and (Rp) for p ∈ (1, n), but a simple argument shows that
(RHp) is not satisfied for p > 2 (we leave it to the reader to check this).
1We remark that the assumption (DG) allows to define e−tLas an operator from L∞(X, µ) to
Theorem 1.1 will be proved by an extensive use of harmonic functions and a fun-damental lemma, which allows to locally “approximate” (in some sense) a function by a harmonic function, see Lemma 2.10.
With similar methods, we can then also give a partial answer to the following question.
Question B. Suppose p ∈ [2, pL). Under which condition does (G
p) imply (Rp+ε)
for some ε > 0?
It is clear that in general, (Gp) cannot imply (Rp+ε) without any additional
assumption. A counterexample is again the example of the two glued copies of Rn.
For n = 2, it is known that for every p > 2, (Rp) does not hold, but (R2) and (G2)
hold trivially. In Subsection 3.2, we prove the following sufficient criterion.
Theorem 1.2. Let (X, µ, Γ, L) be as in Section 1.1 with both Γ and L satisfying the conservation property. Assume that L is a non-negative, self-adjoint operator. If (Gp) and (Pp) hold for some p ≥ 2, then there exists ε > 0 such that (Rp+ε) and
(Gp+ε) hold.
Then in the second part (Section 4), we will focus on the situation where the operator L is only sectorial and not necessarily self-adjoint. We show
Theorem 1.3. Let (X, µ, Γ, L) be as in Section 1.1, with the conservation property for L and Γ. Suppose p ∈ (2, pL). If (G
p) and (Pp) hold, then (Rq) holds for every
q ∈ (2, p).
In such a situation, it is unknown if (Gp) together with (Pp) implies (P2). The
proof of this statement in [12, Theorem 6.3] relies on the self-adjointness of the operator, which is used through the finite propagation speed property to deduce a perfectly localised Caccioppoli inequality. So as far as we know, the assumptions are actually weaker than those in [7].
In the last section, we finally show that reverse H¨older inequalities in Lp are stable under small elliptic perturbations for p close to 2.
Theorem 1.4. Let (M, µ) be a doubling Riemannian manifold with ∇ the Rie-mannian gradient and ∆ the Laplace-Beltrami operator. Assume that the heat ker-nel of (et∆)t>0 satisfies pointwise Gaussian estimates. If the reverse H¨older property
(RHq) holds for −∆ and for some q > 2, then there exists ε > 0 such that for every
p ∈ (2, 2 + ε) and every map A with kA − Idk∞≤ ε and LA= −∇∗A∇ self-adjoint,
the properties (RHp) and (Rp) are satisfied with respect to the operator LA.
2. Examples of Applications and Preliminaries 2.1. Examples and Applications.
• Dirichlet forms and subdomains.
Let (M, d, ν) be a complete space of homogeneous type as above. Consider a self-adjoint operator L on L2(M, ν) and consider E the quadratic form associated with L, that is
E(f, g) = Z
M
If E is a strongly local and regular Dirichlet form (see [24, 29] for precise definitions) with a carr´e du champ structure, then with Γ being equal to this carr´e du champ operator, (R2) and (DG) hold. In particular, this is the
case if (M, d, ν) is a Riemannian manifold with L its non-negative Laplace-Beltrami operator and Γ the length of the Riemannian gradient.
We consider the two following situations:
(a) Consider the global situation (X, d, µ, L) = (M, d, ν, L). In such a case, the typical upper estimate for the kernel of the semigroup is
(DUE) pt(x, y) .
1 q
V (x,√t)V (y,√t)
, ∀ t > 0, a.e. x, y ∈ X.
Under (VD), (DUE) self-improves into a Gaussian upper estimate
(UE) pt(x, y) . 1 V (x,√t)exp −cd 2(x, y) t , ∀ t > 0, a.e. x, y ∈ X.
See [28, Theorem 1.1] for the Riemannian case, [22, Section 4.2] for a metric measure space setting. This yields (pL, pL) = (1, ∞), and the
Faber-Krahn inequality (1.1) holds.
In such a situation, the operator L satisfies the conservation property [29], and the space (X, d, µ) may or may not satisfies a Poincar´e in-equality (Pp).
(b) Local situation with Dirichlet boundary conditions. Given an open sub-set U ⊂ M , we can also look at the heat semigroup in U with Dirichlet
boundary conditions. So consider LU the operator with domain
DU := {f ∈ D(L), supp(f ) ⊂ U } .
Then it is known that LU generates a semigroup in L2(U, ν).
However, it is important to emphasise that due to the Dirichlet bound-ary conditions, the conservation property does not hold for LU!
If the domain U is assumed to be Lipschitz, then under (DUE) for the underlying space, it is known that the semigroup associated with
LU has also a heat kernel with Gaussian pointwise bounds and so
(pLU, p
LU) = (1, ∞). Moreover the (eventual) Poincar´e inequality (P
2)
on the ambiant space (M, d, ν) can be used to study the balls inter-secting the boundary ∂U (as done in e.g. [27, 40]). In particular, it is proved that if (M, d, ν) satisfies (P2) and U is Lipschitz, then for some
p > 2 a reverse H¨older property (RHp) holds for L = LU, as well as
(Gp) and (Rp).
Assume that (M, d, ν, L) is doubling and satisfies a Poincar´e inequality
(P2) with (DUE). Consider U ⊂ M an unbounded inner uniform open
subset (see [29] for a precise definition), which includes the case of Lipschitz domains. Then [29, Theorem 3.12] shows that U equipped with its inner structure is also doubling. In order to get around the fact that LU is not conservative, we can use the h-transform (as described
of Lh = 0 with Dirichlet conditions) and consider the operator Lh Uf = h −1L U(hf ) with domain D[Lh U] := {f ∈ L 2(U, h2dµ), hf ∈ D U}.
Then LhU is self-adjoint with respect to the measure h2dµ and satisfies the conservation property. According to [29], this structure defines a strongly local and regular Dirichlet form with a Poincar´e inequality (P2)
with respect to its carr´e du champ Γ := |∇|. Moreover the semigroup as-sociated with Lh
U has also a heat kernel with Gaussian pointwise bounds
(relatively to the measure h2dµ) so that (p Lh
U, p
Lh
U) = (1, ∞) (see [29]). So by our Theorem 3.6, there exists ε > 0 such that (Rp) holds for
p ∈ [2, 2 + ε) which means Z U |∇f |ph2dµ 1/p . Z U |h−1LU1/2(hf )|ph2dµ 1/p , which corresponds to the Lp(U, h2dµ)-boundedness of ∇h−1L
U1/2(h·).
Moreover, we also have (RHp) for p > 2 close to 2.
• Schr¨odinger operators.
If (M, d, µ) is a doubling Riemannian manifold with ∇ the Riemannian
gradient, we may consider the Schr¨odinger operator L := −∇∗∇ + V
as-sociated with a potential V . Let us focus on the case V ≥ 0, otherwise the situation is more difficult and we refer the reader to [2, 3] (and refer-ences therein) for some works giving assumptions on V to guarantee the boundedness of the Riesz transform. If V is non-negative and belongs to
some Muckenhoupt reverse H¨older space, then boundedness of the Riesz
transform has been obtained in [40, 5]. For V ≥ 0, we may consider Γ given by
Γ(f ) = |∇f |2 + V |f |21/2 or Γ(f ) = |∇f |.
Moreover, if (DUE) holds on the heat semigroup generated by −∇∗∇, then
it also holds for L, hence (pL, pL) = (1, ∞).
Because of the potential V , the operator L does not satisfies the conserva-tion property. If (M, d, µ) satisfies the Poincar´e inequality (P2) with respect
to ∇, then it can be proved that L satisfies a (RHp) property for some p > 2
and with Γ = |∇| (see Proposition 3.10).
As an application of Theorem 3.14, we deduce the following result. Theorem 2.1. Assume that (M, d, µ) is a doubling Riemannian manifold
satisfying (DUE) and (P2). Then for every potential V ≥ 0 there exists
ε > 0 such that for L = −∇∗∇ + V and Γ = |∇|, we have (RHq) as well as
(Gq) and (Rq) for every q ∈ (2, 2 + ε).
It has to be compared with [18] where a negative answer for (Gp) (and
so (Rp)) is given for p > ν if the operator L has a positive ground state
• Second order divergence form operators.
Consider (M, d, µ) a doubling Riemannian manifold (satisfying (VDν)),
equipped with the Riemannian gradient ∇ and its divergence operator div =
∇∗. To A = A(x) a complex matrix-valued function, defined on M and
satisfying the ellipticity (or accretivity) condition (see Section 5 for details), we may define a second order divergence form operator
L = LAf := −div(A∇f ).
Then L is sectorial and satisfies the conservation property but may not be self-adjoint.
In this case, the reverse H¨older property (RHp) for some p > 2 is implied
by (P2), see [27, Chapter 5, Thm 2.1]. We refer the reader to Section 5
where a stability result (with respect to the map A) is shown for the reverse H¨older property.
• Examples where (Gp) is known.
We shall give here two examples of situations where (Gp) is known for some
p > 2. However (Rp) is still unknown, which motivates to ask Question A.
(a) Consider V a non-negative potential on R and define on R2 (equipped
with its Euclidean structure), the operator for x = (x1, x2) ∈ R2
Lf (x) = −∆f (x) + V (x2)f (x).
Then L generates a semigroup, which has a heat kernel satisfying Gauss-ian bounds. For Γ = ∂x1, since V is very nice along the first coordinate, it is natural to ask for the boundedness of ΓL−1/2. Such a
bounded-ness is unknown and does not seem to be trivial, since L−1/2 makes
interact the action of L on the two coordinates. However, it is sur-prising to observe that (Gp) easily holds for every p ∈ [1, ∞]. Indeed,
since L1 := −∂x21 and L2 := −∂
2
x2 + V (x2) commute, we then have e−tL= e−tL1 ⊗ e−tL2 such that k√tΓe−tLkp→p ≤ k √ t∂x1e −tL1k p→pke−tL2kp→p
which is uniformly (with respect to t) bounded. (b) On Rn, consider a second order operator
L = X
|α|≤2
cα(x)∂α
given by bounded measurable complex coefficients cα, depending on
x. For some q > 2, assume that the Lq domain of −L is the
clas-sical Sobolev space Dq(−L) = W2,q(Rn) and also is the generator of
an analytic semigroup (e−tL)t>0 in Lq. Then [35, Theorem 3.1] shows
that a local version of (Gq) holds (with Γ = ∇). In this context, Lq
-boundedness of some local Riesz transforms is unknown.
2.2. Preliminaries. We start by recalling some technical results, which will be needed later.
We denote by M the Hardy-Littlewood maximal operator, defined for f ∈ L1loc(X, µ) and x ∈ X by (2.1) Mf (x) := sup B3x − Z B |f | dµ,
where the supremum is taken over all balls B ⊂ X with x ∈ B. For p ∈ [1, +∞), we abbreviate by Mp the operator defined by Mp(f ) := [M(|f |p)]1/p, f ∈ L1loc(X, µ).
Note that M is bounded in Lq(X, µ) for all q ∈ (1, +∞], cf. [19, Chapitre III].
Consequently, Mp is bounded in Lq(X, µ) for all q ∈ (p, +∞].
Let us recall Gehring’s result [25] which describes a self-improvement of inequal-ities involving maximal functions.
Theorem 2.2. Let (X, d, µ) be a doubling space. Let 1 ≤ p < q < ∞ be two exponents and f ∈ Lqloc(X, µ) such that for some constant C > 0, 1 < α < β < 2 and a ball B we have for almost every x ∈ 2B
sup x∈Q⊂2Q⊂αB − Z Q |f |qdµ 1/q ≤ C sup x∈Q⊂βB − Z Q |f |pdµ 1/p . Then there exists ε := ε(C, p, q, α, β, ν) > 0 such that f ∈ Lq+ε(B) and
− Z B |f |q+εdµ 1/(q+ε) . − Z 2B |f |pdµ 1/p .
The original result is due to Gehring [25, Lemma 2], with a modification in [26, Appendix] which emphasises the restriction to sub-balls Q instead of global maximal functions. There is a large amount of literature on the topic, with extensions to various settings ([27, 32] and also [36]). The proof relies on a suitable Calder´ on-Zygmund decomposition.
Let us also recall the following equivalence between weak and strong Poincar´e inequalities (combining [30, Theorem 3.1] and [34]).
Theorem 2.3. Let (X, µ, Γ, L) be as in Section 1.1. Let p ∈ (1, ∞) and let f ∈ F . The (strong) Lp Poincar´e inequality (Pp) for f is equivalent to the weak version:
there exists λ > 1 such that (w-Pp) − Z B |f − − Z B f dµ|pdµ 1/p . r − Z λB |Γf |pdµ 1/p , where B ranges over balls in X of radius r.
We also recall the following (well-known) fact about Poincar´e inequality (see for instance [30, Theorem 5.1], [23, Theorem 2.7] for similar statements).
Lemma 2.4. Let (X, d, µ) be a doubling metric measure space with (VDν).
As-sume that (Pp) holds for some 1 ≤ p < +∞. Then if q ∈ (p, +∞) is such that
ν1p − 1 q
≤ 1, the following Sobolev-Poincar´e inequality holds: (Pp,q) − Z Br |f − − Z Br f dµ|qdµ 1/q . r − Z Br |Γf |pdµ 1/p , for all f ∈ F , r > 0, and all balls Br with radius r.
We will use the following extrapolation result ([7], [10, Theorem 3.13]).
Proposition 2.5. Let (X, d, µ) be a doubling metric measure space. Let (At)t>0 be
a family of linear operators, uniformly bounded in L2(X, µ). Let T be a sublinear
operator which is bounded on L2(X, µ). Assume that for some q ∈ (2, +∞), every
ball B of radius r > 0 and every f ∈ L2(X, µ), we have
• L2-L2 estimates of T (I − A r): (2.2) − Z B |T (I − Ar)f |2dµ 1/2 . inf x∈BM2(f )(x); • L2-Lq estimates of T (A r): (2.3) − Z B |T Arf |qdµ 1/q . inf x∈B[M2(T f )(x) + M2(f )(x)].
Then, for every p ∈ (2, q), T is bounded on Lp(X, µ).
Let us state a technical result, which describes how a higher order of cancellation with respect to the operator L allows us to gain integrability through off-diagonal estimates.
Lemma 2.6. Let p ∈ [2, pL), and let K > ν
2p. Then for every ball B of radius r > 0
and every function f ∈ L2(X, µ), we have − Z B |(r2L)Ke−r2L f |pdµ 1/p . sup Q⊃B − Z Q |f |2dµ 1/2 , where the supremum is taken over all balls Q containing B.
Proof. We write K = M − ρ with M ≥ 1 an integer and ρ ∈ (0, 1]. Let B be a ball of radius r and f ∈ L2(X, µ). Using a Calder´on reproducing formula, we can write
(2.4) (r2L)M −ρe−r2Lf = c Z ∞ 0 r2 s M −ρ (sL)Me−(s+r2)Lf ds s .
Now recall by analyticity of the semigroup, (tL)Me−tL also satisfies L2-Lp off-diagonal estimates. This yields that for s ≤ r2, the operator (r2L)Me−(s+r2)L satisfies L2-Lp off-diagonal estimates at the scale r, so
− Z B |(r2L)Me−(s+r2)L f |pdµ 1/p . sup Q⊃B − Z Q |f |2dµ 1/2 . Consequently, − Z B | Z r2 0 r2 s M −ρ (sL)Me−(s+r2)Lf |pdµ !1/p . Z r2 0 s r2 ρ − Z B |(r2L)Me−(s+r2)L f |pdµ 1/p ds s . supQ⊃B − Z Q |f |2dµ 1/2 .
For s ≥ r2 on the other hand, (sL)Me−(s+r2)L
satisfies L2-Lp off-diagonal estimates
at the scale s ≥ r2. Denoting by ˜B = √s
r B ⊃ B the dilated ball, we obtain in this
case − Z B |(sL)Me−(s+r2)L f |pdµ 1/p . √s r νp − Z ˜ B |(sL)Me−(s+r2)L f |pdµ 1/p . √s r νp sup Q⊃ ˜B − Z Q |f |2dµ 1/2 . √s r νp sup Q⊃B − Z Q |f |2dµ 1/2 . This gives for M > ν
2p + ρ Z ∞ r2 r2 s M −ρ − Z B |(sL)Me−(s+r2)L f |pdµ 1/p ds s . supQ⊃B − Z Q |f |2dµ 1/2 .
Putting the two parts together yields the conclusion.
2.3. Harmonic functions. Let us first rigorously describe what we mean by har-monic functions. First note that the map
E(f, g) := hLf, gi
is a sesquilinear form defined on D(L). If L is self-adjoint, then it can be extended on D(L1/2) and so in particular in F .
Definition 2.7. Let u ∈ D(L1/2) and B be a ball. We say that u is harmonic on
B if for every φ ∈ D(L1/2) supported on B, one has
E(u, φ) = 0.
Definition 2.8. Suppose p0 ∈ (2, ∞). We say that the reverse H¨older property
(RHp0) holds if for every ball B of radius r > 0 and every function u ∈ F which is harmonic on 2B, one has
(RHp0) − Z B |Γu|p0dµ 1/p0 . − Z 2B |Γu|2dµ 1/2 .
Remark 2.9. By the self-improving of the RHS exponent of a reverse H¨older
in-equality (see [12, Appendix B]), we deduce that a reverse H¨older property (RHp0) for some p0 > 2 implies a L1-Lp0 reverse H¨older property: for every ball B of radius
r > 0 and every function u ∈ F which is harmonic on 2B, one has − Z B |Γu|p0dµ 1/p0 . − Z 2B |Γu| dµ .
First of all, harmonic functions will play a crucial role, so let us detail the main tool which allows us to approximate a function with a harmonic function.
Lemma 2.10. Let (X, µ, Γ, L) be as in Section 1.1 with L non-negative and self-adjoint. Let f ∈ D and consider an open ball B ⊂ X. Then there exists u ∈ F
such that f − u ∈ F is supported in the ball B and u is harmonic in B. Moreover, we have − Z B |Γ(f − u)|2dµ 1/2 . r − Z B |Lf |2dµ 1/2 − Z B |Γu|2dµ 1/2 . − Z B |Γf |2dµ 1/2 + r − Z B |Lf |2dµ 1/2 . (2.5)
We follow the same scheme as in [12, Lemma 4.6], where a proof was given in the particular setting of a Dirichlet form. We explain here how can we extend it to the current more general framework.
Proof. Consider the space of functions
H :=φ ∈ D(L1/2
) ⊂ L2, supp(φ) ⊂ B ⊂ F.
Then, due to (1.1) and the local character of operator Γ, the application φ 7→ kφkH:= kL1/2φkL2 & kΓφkL2(B)
defines a norm on H. Consequently, H equipped with this norm is a Hilbert space, with the scalar product
hφ1, φ2iH := E (φ1, φ2).
Since f ∈ D, the linear form
γ : φ 7→ hLf, φi is continuous on H. Indeed, we have by (1.1)
|γ(φ)| = |hLf, φi| . r Z B |Lf |2dµ 1/2 kΓφkL2(B).
By the representation theorem of Riesz, there exists v ∈ H such that for every φ ∈ H Z B √ Lf√Lφ dµ = E (v, φ) = Z √ Lv√Lφ dµ.
We set u := f − v so that v = f − u being in H is supported in B. Moreover for every φ ∈ H, φ is supported in B so the previous equality yields
E(u, φ) = Z √ Lf√Lφ dµ − Z √ Lv√Lφ dµ = 0. So u is harmonic in B.
Then observe that since f − u is supported in B and u is harmonic on B by using (R2), we have − Z B |Γ(f − u)|2dµ ' 1 |B|k √ L(f − u)k22 ' 1 |B|E(f − u, f − u) ' 1 |B|E(f, f − u).
Since f − u ∈ F is supported on B, Property (1.1) yields E(f, f − u) . r Z B |Lf |2dµ 1/2 kΓ(f − u)kL2(B),
which gives − Z B |Γ(f − u)|2dµ 1/2 . r − Z B |Lf |2dµ 1/2 and so (2.5).
In the previous result, the last quantity in (2.5) may be removed if we make an extra assumption.
Lemma 2.11. Let (X, µ, Γ, L) be as in Section 1.1 with L non-negative and self-adjoint. Assume in addition that for every f ∈ D, every ball B and every function g ∈ F supported in B
(2.6) |hLf, gi| . kΓfkL2(B)kΓgkL2(B).
Then for every f ∈ D and every an open ball B ⊂ X, there exists u ∈ F such that f − u ∈ F is supported in the ball B and u is harmonic in B, that is, for every φ ∈ F supported on B E(u, φ) = 0. Moreover, we have (2.7) − Z B |Γ(f − u)|2dµ 1/2 + − Z B |Γu|2dµ 1/2 . − Z B |Γf |2dµ 1/2 .
We let the reader check that the exact same proof as the one of Lemma 2.10 holds, except that now we can directly control the linear map γ with
|γ(φ)| = |hLf, φi| . Z B |Γf |2dµ 1/2Z B |Γφ|2dµ 1/2 . Z B |Γf |2dµ 1/2 kφkH.
Remark 2.12. The new assumption (2.6) can be thought of as a sharp localised reverse Riesz inequality in L2(M, µ). It is in particular satisfied for second order
elliptic operators L of divergence form, with Γ being the length of the gradient. 3. Boundedness of Riesz transforms of self-adjoint operators
under reverse H¨older property
In this section, (X, µ, Γ, L) will be as in Section 1.1, with the additional assump-tion that L is self-adjoint in L2(X, µ).
3.1. About finite speed of propagation. We first need some technical results on specific approximation operators having finite propagation speed. We recall that for a non-negative, self-adjoint operator L on L2(X, µ), Davies-Gaffney estimates
(DG) for the heat semigroup are equivalent to the fact that the solution of the corresponding wave equation satisfies the finite propagation speed property. See e.g. [38] and [22, Section 3]. The self-adjointness allows us to work with a better functional calculus than just H∞ functional calculus, namely functional calculus based on the Fourier transform. As investigated in [9], this calculus interacts nicely with the finite propagation speed, and yields sharper off-diagonal estimates for operators generated by the calculus.
Lemma 3.1. For every even function ϕ ∈ S(R) with supp ˆϕ ⊆ [−1, 1] and every r > 0, the operator ϕ(r√L) propagates at distance at most r. That is, for all f ∈ L2(X, µ) with supp f ⊆ E ⊆ X, one has
supp(ϕ(r√L)f ) ⊆ {x ∈ X : d({x}, E) ≤ r}.
Moreover we have the following L2 off-diagonal estimates: for all Borel sets E, F ⊆ X, we have (3.1) kϕ(r√L)kL2(E)→L2(F ) . max 1 −d(E, F ) r , 0 , and (3.2) krΓϕ(r√L)kL2(E)→L2(F ). max 1 − d(E, F ) r , 0 .
Assume in addition (Gp0) for some p0 ∈ (2, ∞]. Then for every p ∈ (2, p0), we have L2-Lp off-diagonal estimates: for every pair of balls B
1, B2 of radius r > 0, we have krΓϕ(r√L)kL2(B 1)→Lp(B2) . |B1| 1 p− 1 2 max 1 − d(B1, B2) r , 0 .
Proof. The first statement on ϕ(r√L) and the estimate (3.1) follow from the Fourier inversion formula and the bounded Borel functional calculus of √L, see [9, Lemma 4.4]. The estimate (3.2) then follows from (3.1), the assumption (G2) and the fact
that Γ is supposed to be local. For the proof of L2-Lp off-diagonal estimates for
ϕ(r√L), we use that thanks to real time off-diagonal estimates and the analyticity of (e−tL)t>0 in Lq(X, µ), q ∈ (pL, pL), we have complex time L2-Lp off-diagonal
estimates for the semigroup. The L2-Lp estimates on ϕ(r√L) then again follow
from the Fourier inversion formula. Now assume (Gp0) for some p0 ∈ (2, ∞]. Under this assumption, one has complex time L2-Lp estimates for (√tΓe−tL)
t>0, see e.g.
[4, Proposition 3.16] for a proof. One can once more apply the Fourier inversion
formula, and obtains the last estimate of the lemma.
We may now prove the following version of Lemma 2.6.
Lemma 3.2. Suppose pL > ν, and let p ∈ [2, pL). Let ϕ ∈ S(R) be an even
function with supp ˆϕ ⊆ [−1, 1]. Then for every ball B of radius r > 0 and every function f ∈ L2(X, µ), we have − Z B |r√Lϕ(r√L)f |2dµ 1/2 . sup Q⊃B − Z Q |f |2dµ 1/2 , where the supremum is taken over all balls Q containing B.
Proof. Let p ∈ (ν, pL). Write for M ∈ N with M > ν
zϕ(z) = z(1 + z2)−M(1 + z2)Mϕ(z).
Note that under our assumptions on ϕ, for every K ∈ N the function z 7→ z2Kϕ(z) ∈
S(R) is even with compact Fourier support. We can therefore apply Lemma 3.1 to (1+z2)Mϕ(z) and get L2-L2off-diagonal estimates for (I +r2L)Mϕ(r√L) of the form (3.1). On the other hand, we know that with (I + r2L)−1 also (r2L)1/2(I + r2L)−M
satisfies L2-Lp off-diagonal estimates of order N := 1 2 + ν 2( 1 2 − 1 p), see e.g. [8,
Proposition 5.3]. Combining the two estimates gives L2-Lp off-diagonal estimates of order N for r√Lϕ(r√L). We therefore get
− Z B |r√Lϕ(r√L)f |2dµ 1/2 ≤ − Z B |r√Lϕ(r√L)f |pdµ 1/p .X j≥0 2−2jN|B|−1/2 Z B |f |2dµ 1/2 .X j≥0 2−j(2N −ν2) − Z 2jB |f |2dµ 1/2 . sup Q⊃B − Z Q |f |2dµ 1/2 ,
where we used N > ν4 (since p > ν) in the last estimate.
3.2. Poincar´e inequalities and gradient estimates.
Proposition 3.3 (Lp Caccioppoli inequality). Let (X, µ, Γ, L) as in Section 1.1 with L self-adjoint. Assume (Gp) for some p ∈ [2, +∞]. Then for every q ∈ (pL, p]
and every integer N ,
(3.3) r − Z Br |Γf |qdµ 1/q . − Z 2Br |f |qdµ 1/q + − Z 2Br |(r2L)Nf |qdµ 1/q
for all f ∈ D(LN) and all balls Br of radius r.
The result was shown in [12, Proposition 5.4] in the case N = 1. The proof for general N ∈ N is similar, we give a proof here for the sake of completeness.
Proof. Consider an even function ϕ ∈ S(R) with supp ˆϕ ⊆ [−1, 1] and ϕ(0) = 1.
Consequently, ϕ0(0) = 0, and z 7→ z−1ϕ0(z) ∈ S(R) is even with Fourier support in [−1, 1], cf. [9, Lemma 6.1]. Fix a ball B of radius r > 0, an exponent q ∈ (1, p] and split
f = ϕ(r√L)f + (I − ϕ(r√L))f. Since ϕ(0) = 1, one has
(I − ϕ(r√L)) = Z r
0
√
Lϕ0(s√L) ds.
Using the finite propagation speed property applied to the functions ϕ and z 7→ z−1+2Nϕ0(z), we have that both ϕ(r√L) and (r2L)−N(1 − ϕ(r√L)) satisfy the
propagation property at a speed 1 and so propagate at a distance at most r. The same stills holds by composing with the gradient. Hence,
kΓf kLq(B) . kΓϕ(r √
L)kq→qkf kLq(2B)
+ kΓ(1 − ϕ(r√L))(r2L)−Nkq→qk(r2L)Nf kLq(2B). (3.4)
For q ∈ (2, p], one obtains (Gq) via interpolation between (G2) and (Gp). For
transform as (1 + r2L)−1 = R0+∞e−t(1+r2L)dt, we deduce gradient bounds for the resolvent in Lq, that is kΓ(1 + r2L)−1k q→q . Z +∞ 0 e−tkΓe−tr2Lk q→qdt . Z +∞ 0 e−t r√tdt . r −1 .
Denote ψ := ϕ or ψ := x 7→ (1 − ϕ(x))/x2N, and consider λ(x) = ψ(x)(1 + x2). Hence ψ(r√L) = (1 + r2L)−1λ(r√L) and therefore kΓψ(r√L)kq→q ≤ kΓ(1 + r2L)−1kq→qkλ(r √ L)kq→q . r−1kλ(r √ L)kq→q.
Observe that λ ∈ S(R) since ϕ(0) = 1 and ϕ0(0) = 0. By a functional calculus
result (see e.g. [13]), we then have sup
r>0
kλ(r√L)kq→q . 1,
and consequently,
kΓψ(r√L)kq→q . r−1.
Coming back to (3.4), we obtain
kΓf kLq(B) . r−1kf kLq(2B)+ rkLf kLq(2B).
We are now going to give a more general version of the main result of [6]. More precisely, [6, Theorem 0.4] states that if X is a complete non-compact Riemannian manifold satisfying the doubling condition (VD) and L is its nonnegative Laplace-Beltrami operator, then under the Poincar´e inequality (P2) there exists ε > 0 such
that (Rp) holds for p ∈ [2, 2 + ε). This result relies on the self-improvement of
Poincar´e inequalities from [34], but also on considerations on the Hodge projector that are specific to the Riemannian setting. We give here a proof that is valid in our more general setting and also gives a Lp0-version.
Theorem 3.4. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint and L and Γ satisfying the conservation property. If for some p0 ∈ [2, ∞), the combination (Pp0) and (Gp0) holds, then there exists ε > 0 such that (Gp) holds for p ∈ [p0, p0+ ε). Proof. Let N ∈ N to be chosen later. Consider f ∈ Lp0(X, µ), t > 0 and set g := e−tLf . For every ball Br with radius r > 0, the Caccioppoli inequality (3.3)
together with (VD) yields − Z Br |Γg|p0dµ 1/p0 . 1 r − Z 2Br g − − Z 2Br g dµ p0 dµ 1/p0 +r−1 − Z 2Br |(r2L)Ng|p0dµ 1/p0 . We know by [34] that (Pp0) implies (Pp0−κ) for κ > 0 small enough which self-improves into a Sobolev-Poincar´e inequality (Pp0,p0−κ) (see Lemma 2.4). Hence, for κ small enough − Z 2Br g − − Z 2Br g dµ p0 dµ 1/p0 . r − Z 2Br |Γg|p0−κdµ 1/(p0−κ) .
We obtain in this way (3.5) − Z Br |Γg|p0dµ 1/p0 . − Z 2Br |Γg|p0−κdµ 1/(p0−κ) + r−1 − Z 2Br |(r2L)Ng|p0dµ 1/p0 . Now, for a ball B√
tof radius
√
t and a parameter K > ν to be chosen later, consider the function h := |Γg| + c(f, B√ t), where c(f, B√ t) := t −1/2 inf x∈B√ t M2(f )(x)
is a constant. Since g = e−tLf , for N sufficiently large Lemma 2.6 implies that for every ball Br ⊂ B√t with radius r,
r−1 − Z 2Br |(r2L)Ne−tL f |p0dµ 1/p0 . c(f, B√ t). Therefore (3.5) yields − Z Br |Γg|p0dµ 1/p0 . − Z 2Br |Γg|p0−κdµ 1/(p0−κ) + c(f, B√ t). (3.6) It follows that (3.7) − Z Br hp0dµ 1/p0 . − Z 2Br hp0−κdµ 1/(p0−κ) . Now for x ∈ B√
t consider the quantities
Fp0(x) := sup x∈B⊂2B√ t − Z B hp0dµ 1/p0 and Fp0−κ(x) := sup x∈B⊂2B√ t − Z B hp0−κdµ 1/(p0−κ) ,
where the supremum is taken over all the balls B containing x and included into B√
t. Then, we have
Fp0(x) . Fp0−κ(x).
Indeed, for B ⊂ 2B√
t a ball containing x, if 2B ⊂ 2B√t then we may apply (3.7),
and if 2B is not included in 2B√
t (since x ∈ B ∩ B√t) then r(B) '
√
t and so we directly apply off-diagonal estimates to have
− Z Br hp0dµ 1/p0 . c(f, B).
We may apply Gehring’s Lemma (Theorem 2.2), and we have for some ε > 0 that − Z B√ t |h|pdµ !1/p . − Z 2B√ t |h|p0dµ !1/p0
for every p ∈ (p0, p0+ ε). Hence, with (3.7) applied for r = √ t, it follows that (3.8) − Z B√ t |h|pdµ !1/p . − Z 4B√ t |h|p0−κdµ !1/(p0−κ) . Hence, we deduce that
− Z B√ t |Γe−tLf |pdµ !1/p . − Z 4B√ t |Γe−tLf |p0−κdµ !1/(p0−κ) + c(f, B√ t).
Interpolating the Davies-Gaffney estimates (DG) with (Gp0) yields L
2− Lp0−κ off-diagonal estimates for √tΓe−tL and so for a large enough integer d ≥ 1
− Z B√ t |√tΓe−tLf |pdµ !1/p . inf x∈B√ t M2(f )(x) . − Z B√ t M2(f )pdµ !1/p . By summing over a covering of X by balls with radius√t with the Lp-boundedness
of the maximal function, we then deduce that√tΓe−tLis bounded on Lp, uniformly
with respect to t > 0, which is (Gp) as desired.
Remark 3.5. Using the self-improvement of reverse H¨older inequality [12, Appen-dix B], (3.7) yields for any ρ ∈ (1, 2)
− Z Br hp0dµ 1/p0 . − Z 2Br hρdµ 1/ρ .
So a careful examination of the previous proof shows that we have in fact the fol-lowing inequality: for every ball Br of radius r > 0 and every g ∈ D(LN),
− Z Br |Γg|p0dµ 1/p0 . − Z 2Br |Γg|ρdµ 1/ρ + r−1k(r2L)Ngk L∞(4B r).
By combining Theorem 4.3 (which will be proved later in a more general setting with a sectorial operator L) and Theorem 3.4 with the fact that (Rp) always implies
(Gp) by Lp analyticity of the semigroup and the fact that the Poincar´e inequality
is weaker and weaker as p increases, we deduce the following statement, which encompasses Theorems 4.3 and 3.4 and extends both [7, Theorem 1.3] and [6, Theorem 0.4].
Theorem 3.6. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint and Γ and L satisfying the conservation property. If for some p0 ∈ [2, ∞], the combination
(Pp0) with (Gp0) holds, then there exists p1 ∈ (p0, ∞] such that
(pL, p1) = {p ∈ (pL, ∞), (Gp) holds} = {p ∈ (pL, ∞), (Rp) holds}.
3.3. Reverse H¨older property and Poincar´e inequalities. First let us recall
the following result about L2 Poincar´e inequalities, proved in [12, Theorem 6.1] combined with Theorem 2.3.
Theorem 3.7. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint. Suppose that for every ball B of radius r > 0 and every function u ∈ F harmonic on 2B,
we have − Z B |u − − Z B u dµ|2dµ 1/2 . r − Z B |Γu|2dµ 1/2 . Then the Poincar´e inequality (P2) holds.
From this result we obtain
Corollary 3.8. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint. Assume (RHp0) for some p0 ∈ (2, ∞). Then (Pp) for some p ∈ (2, p0] implies (P2).
Proof. By (Pp) which implies (Pp0), we have for every ball B of radius r > 0 and every function u ∈ F which is harmonic on 2B
− Z B u − − Z B u dµ p0 dµ 1/p0 . r − Z B |Γu|p0dµ 1/p0 . So we deduce with (RHp0) that
− Z B u − − Z B u dµ 2 dµ !1/2 ≤ − Z B u − − Z B u dµ p0 dµ 1/p0 . r − Z B |Γu|p0dµ 1/p0 . r − Z 2B |Γu|2dµ 1/2 .
By Theorem 2.3, we obtain (P2) for every harmonic function. The full Poincar´e
inequality (P2) then follows by Theorem 3.7.
Proposition 3.9. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint and Γ and L satisfying the conservation property. Assume (Gp0) for some p0 ∈ (2, ∞). Then (Pp0) yields (RHp0).
Proof. Let B be a ball of radius r > 0, and u ∈ D harmonic on 2B. Then Proposi-tion 3.3 yields that (Gp0) implies a L
p0-Caccioppoli inequality, so using the conser-vation, it follows that
− Z B |Γu|p0dµ 1/p0 . 1 r − Z 2B u − − Z 2B u dµ p0 dµ 1/p0 .
By the self-improvement of Poincar´e inequalities (see [34]), there exists ε > 0 such that (Pp0−ε) holds and so for a small enough ε > 0 the Sobolev-Poincar´e inequality (Lemma 2.4) yields − Z B |Γu|p0dµ 1/p0 . − Z 2B |Γu|p0−εdµ 1/(p0−ε) .
Then the exponent of the RHS of such a reverse H¨older inequality can always be improved (see [12, Theorem B.1]), so that we deduce
− Z B |Γu|p0dµ 1/p0 . − Z 2B |Γu|2dµ 1/2 ,
In the Riemannian setting with L denoting the Laplace-Beltrami operator, it is already known that the Poincar´e inequality (P2) implies a reverse H¨older property
(RHp) for exponents p > 2 sufficiently close to 2, see [6, Proposition 2.2]. We
extend this result here (with a slightly different proof) for the sake of completeness. We are considering the setting of Dirichlet forms with a carr´e du champ Γ (see the first example of Subsection 2.1), see also [29] and [12] for details.
Proposition 3.10. Let (X, µ, Γ, L) be as in Section 1.1, and assume that L gener-ates a strongly local and regular Dirichlet form such that (X, d, µ, E ) is a doubling metric measure Dirichlet space with a carr´e du champ Γ and that (1.1) and (P2)
are satisfied. Then there exists ε > 0 such that (RHp) holds for every p ∈ (2, 2 + ε).
Remark 3.11. Note that the proof only requires the conservation property for Γ and not for the operator L.
Proof. Let B0 = B(x0, r0) be a ball and f ∈ D be a function, harmonic on 2B0.
Then for every sub-ball B ⊂ B0 of radius r ≤ r0, consider a Lipschitz function χB,
supported on 2B, equal to 1 on B with kΓχBk∞ . r−1. We have (by using the
Dirichlet form structure and Leibniz inequality for carr´e du champ Γ), since χB is
supported on 2B0 where f is harmonic
Z χ2B|Γf |2dµ ≤ Z f − − Z B f dµ χB|ΓχB| |Γf | dµ.
Consider q > 2 such that max(0,12 − 1 ν) <
1 q <
1
2. Then by Lemma 2.4, we know
that (P2) implies a L2-Lq Sobolev-Poincar´e inequality. H¨older’s inequality (with q0
the conjugate exponent of q) yields Z χ2B|Γf |2dµ . Z χqB0|Γf |q0dµ 1/q0 r−1 Z 2B f − − Z B f dµ q dµ 1/q . Using the L2-Lq Sobolev-Poincar´e inequality, this implies
(3.9) Z χ2B|Γf |2 dµ . Z χqB0|Γf |q0 dµ 1/q0Z 2B |Γf |2dµ 1/2 .
Then let us fix a ball B1 ⊂ B0 and consider (Qj)j a Whitney covering of B1, which
is a collection of balls satisfying in particular
(Qj)j is a covering of B1 and (2Qj)j is also a bounded covering of B1.
So (3.9) applied to each Qj implies
Z Qj |Γf |2 dµ . Z 2Qj |Γf |q0dµ 1/q0Z 2Qj |Γf |2dµ 1/2 . By summing over j, Cauchy-Schwarz’s inequality gives
Z B1 |Γf |2dµ . X j Z 2Qj |Γf |q0dµ 2/q0!1/2Z B1 |Γf |2dµ 1/2 . Consequently, since q0 < 2 (so `q0 ⊂ `2), we deduce that
Z B1 |Γf |2dµ 1/2 . Z B1 |Γf |q0dµ 1/q0 .
This property holds for every sub-ball B1 ⊂ B0. We may then apply Gehring’s
argument: there exists ε > 0 such that Z B0 |Γf |pdµ 1/p . Z 2B0 |Γf |2dµ 1/2 ,
uniformly with respect to the ball B1 for every p ∈ (2, 2 + ε).
3.4. Boundedness of Riesz transforms in a self-adjoint setting. Let us now focus on the situation where L does not satisfy the conservation property. In this case, the use of Poincar´e inequalities does not seem to help and we aim to explain how the reverse H¨older inequalities (RHp) can be used to get around this difficulty.
Theorem 3.12. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint. If pL ≤ ν,
assume in addition (2.6). If (Gp0) and (RHp0) hold for some p0 ∈ (2, p
L), then (R p)
is satisfied for every p ∈ (2, p0).
Proof. We first assume pL > ν. We will apply Proposition 2.5 to the Riesz
trans-form. Consider an even function ϕ ∈ S(R) with supp ˆϕ ⊆ [−1, 1] and ϕ = 1 on
[−1/2, 1/2]. Let q ∈ (2, p0). Let f ∈ L2(X, µ), and let B be a ball of radius r > 0.
We first check (2.2) with Ar := ϕ(r
√
L) and T = R. We follow the arguments of [7, Lemma 3.1], adapted to our approximation operators Ar. We use the integral
representation I − ϕ(r √ L) = Z r 0 √ Lϕ0(s √ L) ds to write T (I − ϕ(r √ L))f ≤ Z r 0 Γϕ 0 (s√L)f ds.
Since ϕ0 is supported in [−1, −1/2] ∪ [1/2, 1], we may repeat similar arguments as in [7] to have L2-L2 off-diagonal estimates for Γϕ0(s√L) and then obtain with the
L2-boundedness of T and (3.1) that
− Z B |T (I − ϕ(r√L))f |2dµ 1/2 . inf x∈BM2(f )(x), (3.10)
which concludes the proof of (2.2).
Let us now check (2.3), again with T = R and Ar = ϕ2(r
√
L). Let h :=
ϕ(r√L)L−1/2f . By Lemma 2.11, there exists u ∈ F such that h − u ∈ F is
supported in the ball 2B and u is harmonic in 2B with (3.11) − Z 2B |Γ(h − u)|2dµ 1/2 + − Z 2B |Γu|2dµ 1/2 . − Z 2B |Γh|2dµ 1/2 +r − Z 2B |Lh|2dµ 1/2 . Then we split, with h = ϕ(r√L)L−1/2f ,
|ΓL−1/2ϕ2(r√L)f | = |Γϕ(r√L)h|
≤ |Γϕ(r√L)(h − u)| + |Γ(I − ϕ(r√L))u| + |Γu|. (3.12)
For the first quantity, we use L2-Lp off diagonal estimates of Γϕ(r√L) (see Lemma
3.1) with the support condition of h − u and the Faber-Krahn inequality (1.1) to obtain − Z B Γϕ(r √ L)(h − u) p dµ 1/p . r−1 − Z 2B |h − u|2 dµ 1/2 . − Z 2B |Γ(h − u)|2 dµ 1/2 . − Z 2B |Γh|2 dµ 1/2 + r − Z 2B |Lh|2dµ 1/2 , where we have used (3.11) in the last step.
The second quantity in (3.12) is equal to 0. To see this, write (I − ϕ(r√L))u = Z r 0 √ Lϕ0(s√L)u ds = Z r 0 L−1/2ϕ0(s√L)Lu ds.
Since ϕ is an even function with its spectrum included in [−1, 1], also x 7→ x−1ϕ0(x) is even with its spectrum still included in [−1, 1]. By the finite propagation speed property, we deduce that (sL)−1/2ϕ0(s√L) propagates at a distance at most s ≤ r. Since u is harmonic on 2B then we deduce that
(sL)−1/2ϕ0(s√L)Lu = 0 on B.
Therefore (I − ϕ(r√L))u = 0 on B, and by the locality property of the gradient,
|Γ(I − ϕ(r√L))u| = 0 on B.
It remains to study the last term in (3.12). Using the reverse H¨older property (RHp0) and that u is harmonic on 2B gives
− Z B |Γu|pdµ 1/p . − Z 2B |Γu|2dµ 1/2 . − Z 2B |Γh|2dµ 1/2 + r − Z 2B |Lh|2dµ 1/2 , where we have used again (3.11).
Consequently, we have obtained − Z B |ΓL−1/2ϕ(r√L)f |pdµ 1/p (3.13) . − Z 2B |ΓL−1/2ϕ(r√L)f |2dµ 1/2 + r − Z 2B |√Lϕ(r√L)f |2dµ 1/2 . − Z 2B |ΓL−1/2(ϕ(r√L) − I)f |2dµ 1/2 + − Z 2B |ΓL−1/2f |2dµ 1/2 + − Z 2B |r√Lϕ(r√L)f |2dµ 1/2 .
The first term is bounded by infx∈BM2(f )(x) by (3.10). So by Lemma 3.2 with
the assumption pL> ν, we then conclude
− Z B |ΓL−1/2ϕ(r√L)f |pdµ 1/p . − Z 2B |ΓL−1/2f |2dµ 1/2 + inf x∈BM2(f )(x),
which implies (2.3).
We may then apply Proposition 2.5 to the Riesz transform and conclude the proof of (Rp) for every p ∈ [2, p0).
If otherwise pL ≤ ν, the assumption (2.6) implies that (3.11) can be improved to − Z 2B |Γ(h − u)|2dµ 1/2 + − Z 2B |Γu|2dµ 1/2 . − Z 2B |Γh|2dµ 1/2 . This in turn improves (3.13) to
− Z B |ΓL−1/2ϕ(r√L)f |pdµ 1/p . − Z B |ΓL−1/2ϕ(r√L)f |2dµ 1/2 . − Z B |ΓL−1/2(ϕ(r√L) − I)f |2dµ 1/2 + − Z 2B |ΓL−1/2f |2dµ 1/2 . − Z 2B |ΓL−1/2f |2dµ 1/2 + − Z 4B |f |2dµ 1/2 ,
where in the last step, we use the global L2-boundedness of ΓL−1/2(ϕ(r√L) − I)
and the local property of Γ together with the finite propagation speed property. In the same spirit, we also improve one of the main result of [6].
Proposition 3.13. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint and pL > ν. Assume that for some q > 2 a reverse H¨older property (RH
q) is satisfied.
Then there exists ε > 0 such that (Gp) holds for p ∈ (2, 2 + ε).
We do not detail the proof of this proposition here, since in Section 5 we will prove a stronger result (involving also an elliptic perturbation of the considered operator), see Proposition 5.2. We let the reader check that the proof written there in the context of a Riemannian manifold only relies on Lemma 2.10 and so holds in our more general setting.
By combining Theorem 3.12 with Proposition 3.13, we get the following result. Theorem 3.14. Let (X, µ, Γ, L) be as in Section 1.1 with L self-adjoint and pL> ν.
Assume that for some q > 2 a reverse H¨older property (RHq) is satisfied. Then
there exists ε > 0 such that (Rp) holds for p ∈ (2, 2 + ε).
Remark 3.15. • According to Proposition 3.10, our assumptions are weaker
than the Poincar´e inequality (P2), so the previous result improves [6,
Theo-rem 0.4].
• We emphasise that the proof only relies on Gehring’s result [25] (which can be thought of as an application of a Calder´on-Zygmund decomposition), and never uses the very deep result of Keith-Zhong [34] about the self-improvement of Poincar´e inequalities which was used in [6].
4. Boundedness of Riesz transforms for non self-adjoint operators
without (P2)
Our aim in this section is to improve the understanding of the links between gradient estimates (Gp), boundedness of the Riesz transform (Rp) through Poincar´e
inequalities, as initiated in [7] and [6]. Here, we focus on the case where the operator L is not self-adjoint, but only sectorial.
4.1. Caccioppoli inequality.
Proposition 4.1 (Lp Caccioppoli inequality). Let (X, µ, Γ, L) be as in Section 1.1. Assume (Gp0) for some p0 ∈ (2, p
L). Suppose p ∈ (2, p
0), M ∈ N and N > 0. Then
for every f ∈ D(LM) and every ball B of radius r > 0,
− Z B |rΓf |pdµ 1/p .X `≥0 2−`N " − Z 2`B |f |pdµ 1/p + − Z 2`B |(r2L)Mf |pdµ 1/p# . Suppose in addition M > ν2(12 − 1 p) + 1
2. Then for every f ∈ D(L
M) and every ball
B of radius r > 0, − Z B |rΓf |pdµ 1/p .X `≥0 2−`N " − Z 2`B |f |2dµ 1/2 + − Z 2`B |(r2L)Mf |2dµ 1/2# . Remark 4.2. The second statement is more accurate than the first one. It is interesting to observe that it corresponds to a localised Sobolev embedding with the suitable scaling, since it yields a local version of (with the standard notation for Sobolev spaces) WL2M,2 ,→ WΓ1,p with the usual condition (2M − 1) > ν12 − 1
p
. Proof. Let p ∈ (2, p0). For f ∈ D(LM), write
f = (I − (I − e−r2L)2M)f + (I − e−r2L)2Mf = 2M X k=1 cke−kr 2L f + (r2L)−M(I − e−r2L)2M(r2L)Mf,
where ck are binomial coefficients. The statement of the proposition will then be
a direct consequence of Lp-Lp off-diagonal estimates (L2-Lp off-diagonal estimates,
resp.) at scale r of the operators rΓe−kr2Land rΓ(r2L)−M(I −e−r2L)2M. By interpo-lating (Gp0) with (DG), one obtains L
p-Lp off-diagonal estimates with exponential
decay for the gradient of the semigroup, and in particular for given ˜N > 0
(4.1) krΓe−r2Lf kLp(E)→Lp(F ).
1 + d(E, F ) r
− ˜N
for all r > 0 and all Borel sets E, F ⊂ X. On the other hand, e−r2/2L satisfies L2
-Lp off-diagonal estimates by assumption. Composing these estimates with Lp-Lp
off-diagonal estimates for rΓe−r2/2L yields the desired L2-Lp off-diagonal estimates for rΓe−r2L. For the second operator, one writes
I − e−r2L = − Z r 0 ∂se−s 2L ds = 2(r2L)Sr
with Sr = Z r 0 s r 2 e−s2Lds s , therefore (4.2) (r2L)−M(I − e−r2L)2M = cSrM M X `=0 e−`r2L.
From (4.1), we have that for s ∈ (0, r), sΓe−s2Lsatisfies Lp-Lpoff-diagonal estimates at scale s and therefore also at scale r. Thus,
krΓSrkLp(E)→Lp(F ). Z r 0 s r 2 krΓe−s2LkLp(E)→Lp(F ) ds s . 1 + d(E, F ) r − ˜N .
By composing these off-diagonal estimates with Lp-Lp off-diagonal estimates for
e−r2L, one obtains from (4.2) that the operator rΓ(r2L)−M(I − e−r2L
)2M satisfies
Lp-Lp off-diagonal estimates at scale r. Since ˜N can be chosen arbitrarily, this
already gives the first statement of the proposition. To finish the proof of the second statement, assume now that M > ν2(12−1
p) + 1
2. Let B be a ball with radius
r, let 2 ≤ q1 ≤ q2 ≤ p. Then Lq1-Lq2 off-diagonal estimates for sΓe−s
2L yield − Z B |rΓSrf |q2dµ 1/q2 ≤ Z r 0 s r 2 − Z B |rΓe−s2Lf |q2dµ 1/q2 ds s . ∞ X `=0 2−`N Z r 0 s r 1−ν(1 q1− 1 q2) ds s − Z 2`B |f |q1dµ 1/q1 . ∞ X `=0 2−`N − Z 2`B |f |q1dµ 1/q1 ,
under the assumption that ν(q1 1 −
1
q2) < 1. An analogous computation yields − Z B |Srf |q2dµ 1/q2 . ∞ X `=0 2−`N Z r 0 s r 2−ν(q11 −q21) ds s − Z 2`B |f |q1dµ 1/q1 . ∞ X `=0 2−`N − Z 2`B |f |q1dµ 1/q1 ,
under the assumption that ν(q1 1−
1
q2) < 2. By iterating these off-diagonal estimates,
we obtain that rΓSM
r satisfies L2-Lp off-diagonal estimates. We finally combine
these estimates with L2-L2 off-diagonal estimates for e−r2L, we then get L2-Lp off-diagonal estimates for the operator rΓ(r2L)−M(I −e−r2L
)2Mfrom the representation
4.2. Boundedness of Riesz transform under conservation property. Let us first state an improvement of one the main results in [7]: the point is that we are able to replace (P2) by the weaker assumption (Pp0) for p0 > 2.
Theorem 4.3. Let (X, µ, Γ, L) be as in Section 1.1 with the conservation property for both Γ and L. Assume for some p0 ∈ [2, pL) that the combination (Pp0) with (Gp0) holds. If p0 > 2 then (Rp) holds for every p ∈ (pL, p0). If p0 = 2 then there exists ε > 0 such that (Rp) holds for every p ∈ (pL, 2 + ε).
Remark 4.4. For p0 < ν, where ν is the exponent in (VDν), (Pp0) is not necessary for (Rp) to hold for every p ∈ (1, p0), as the example of the connected sum of two
copies of Rn shows (see [17]).
Remark 4.5. In [12, Theorem 6.3], it is shown that if L is self-adjoint then (Gp)
with (Pp) for some p > 2 implies (P2). This argument goes trough a (perfectly)
localised Caccioppoli inequality. In the case where the operator is not self-adjoint, such an inequality is unknown. Even if we are showing a weaker version with off-diagonal terms (Proposition 4.1), it is not clear how this would allow us to adapt the arguments of [12] in order to obtain (P2) from (Gp) + (Pp).
So as far as we know, Theorem 4.3 is a real improvement with respect to [7]. Proof of Theorem 4.3. We only consider the case p0 > 2. The case p0 = 2 can be
treated similarly. We apply Proposition 2.5 to the Riesz transform R := ΓL−1/2. Let q ∈ (2, p0), and let D ∈ N with D > ν4. Moreover, let f ∈ L2(X, µ) and B be a
ball of radius r > 0. For t > 0, denote
Pt(D):= I − (I − e−tL)D.
We check in two steps the assumptions of Proposition 2.5 for T = R and Ar2 = Pr(D)2 .
Step 1: Checking of (2.2).
Following [7, Lemma 3.1], which only relies on the Davies-Gaffney estimates (DG), we already know that (2.2) is satisfied for T = R and Ar2 = P(D)
r2 . More precisely, it is proven that
(4.3) − Z B |ΓL−1/2(I − Pr(D)2 )f | 2dµ 1/2 . ∞ X j=0 2j(ν2−2D) − Z 2jB |f |2dµ 1/2 , which in particular yields (2.2).
Step 2: Checking of (2.3).
Let us now check (2.3), again with T = R and Ar2 = P(D)
r2 , which is (4.4) − Z B |ΓL−1/2Pr(D)2 f | qdµ 1q . inf x∈B M2(ΓL −1/2 f )(x) + M2(f )(x) .
By Lemma 4.6 and Remark 4.7, applied to g = L−1/2f and M ∈ N with M >
1 2( ν p + 1), we deduce that − Z B |ΓL−1/2Pr(D)2 f | q dµ 1q . − Z 2B |ΓL−1/2Pr(D)2 f | 2 dµ 12 +X `≥0 2−`N − Z 2`B |(r2L)M −1/2e−r2Lf |p0dµ p01 . (4.5)
Let us now estimate the two quantities. We get from (4.3) − Z 2B |ΓL−1/2Pr(D)2 f | 2 dµ 1/2 ≤ − Z 2B |ΓL−1/2f |2dµ 1/2 + − Z 2B |ΓL−1/2(I − Pr(D)2 )f | 2 dµ 1/2 . − Z 2B |ΓL−1/2f |2dµ 1/2 + inf x∈BM2(f )(x) . inf x∈B M2(ΓL −1/2 f )(x) + M2(f )(x) . (4.6)
Moreover, by covering 2`B with approximately 2ν` balls of radius r, we obtain from Lemma 2.6 − Z 2`B |(r2L)M −1/2e−r2 2 Lf |p0dµ p01 . 22ν`sup Q⊃B − Z Q |f |2dµ 1/2 . 22ν`inf x∈B M2(f )(x).
Consequently, plugging these two last estimates into (4.5) with N chosen large enough, we deduce − Z B |ΓL−1/2P(D) r2 f | qdµ 1/q . inf x∈B M2(ΓL −1/2f )(x) + M 2(f )(x) ,
which is (4.4). In this way, we obtain (2.3), and the proof is complete by Proposition 2.5. To be more precise, Proposition 2.5 implies that the Riesz transform is bounded on Lp(X, µ), for every p ∈ (2, q) with arbitrary q ∈ (2, p
0).
Lemma 4.6. Assume the conservation property for L and Γ as well as (Gp0) and
(Pp0) for some p0 ∈ [2, p
L). Suppose N > 0, and M ∈ N with M > 1 2(
ν
p0 + 1). If p0 > 2, suppose p ∈ [2, p0). Then for every g ∈ D(LM), every t > 0 and every ball
B of radius √t, − Z B |√tΓe−tLg|pdµ 1/p . − Z 2B |√tΓe−tLg|2dµ 1/2 +X `≥0 2−`N − Z 2`B |(tL)Me−tL g|p0dµ 1/p0 .
If p0 = 2, then there exists ε > 0 such that the same property holds for every
p ∈ (2, 2 + ε).
Remark 4.7. As we will see in the proof, the operators √tΓe−tL in the LHS and
RHS can be replaced by √tΓPt(D) for any integer D.
Proof. We only consider the case p0 > 2. The case p0 = 2 can be treated similarly
with p0 = p = 2, since we already have Davies-Gaffney estimates (DG) by
conservation property, we then have for every r ∈ (0,√r) and every ball Br ⊂ B − Z Br |Γe−tLg|pdµ 1/p .X `≥0 2−` ˜N " r−1 − Z 2`B r e−tLg − − Z Br e−tLgdµ p dµ 1/p + r−1 − Z 2`B r |(r2L)Me−tL g|pdµ 1/p# ,
where M ∈ N, and ˜N > 0 can be chosen arbitrarily large. Using (Pp0), which self-improves into (Pp0−κ) for κ small enough (see [34]), and Lemma 2.4, we deduce that for κ > 0 small enough and p ∈ (p0− κ, p0), we have
1 r − Z 2`B r e−tLg − − Z Br e−tLg dµ p dµ 1/p . ` X j=0 1 r − Z 2jB r e−tLg − − Z 2jB r e−tLg dµ p dµ 1/p . ` X j=0 2j − Z 2jB r |Γe−tLg|p0−κdµ 1/p0−κ .
We thus obtain, up to changing ˜N to some other parameter N (but which can still be chosen arbitrarily large), for every r ∈ (0,√t) and every ball Br ⊂ B
− Z Br |Γe−tLg|pdµ 1/p .X `≥0 2−`N " − Z 2`B r |Γe−tLg|p0−κdµ 1/(p0−κ) (4.7) + r−1 − Z 2`Br |(r2L)Me−tLg|p0dµ 1/p0# . Now consider the function
h := |Γe−tLg| + c(g, B), where c(g, B) := t−1/2X `≥0 2−`N − Z 2`B |(tL)Me−tL g|p0dµ 1/p0
is a constant. We have by using (VDν) and r ≤
√ t r−1 − Z 2`B r |(r2L)Me−tL g|p0dµ 1/p0 . r √ t 2M −1√ t r ν/p0 t−1/2 − Z 2`B |(tL)Me−tL g|p0dµ 1/p0 . t−1/2 − Z 2`B |(tL)Me−tLg|p0dµ 1/p0 , choosing M > 12(pν 0 + 1). Therefore (4.7) yields − Z Br |Γe−tLg|pdµ 1/p . X `≥0 2−`N − Z 2`B r |Γe−tLg|p0−κdµ 1/(p0−κ) + c(g, B).
Since c(g, B) is a constant (and it is also equal to any of its average), it follows that (4.8) − Z Br hpdµ 1/p .X `≥0 2−`N − Z 2`B r hp0−κdµ 1/(p0−κ) . Now for x ∈ 2B, we consider the quantities
Fp(x) := sup x∈Br⊂2B − Z Br hpdµ 1/p
where the supremum is taken over all the balls Br containing x and included into
2B and Fp0−κ(x) := sup x∈Br − Z Br hp0−κdµ 1/(p0−κ) := Mp0−κ[h](x), We have proved that for every x ∈ 2B
Fp(x) . Fp0−κ(x).
We may then apply Gehring’s Lemma (see Theorem 2.2), and we deduce that for some ε > 0 − Z B |h|p+εdµ 1/(p+ε) . − Z 2B |h|pdµ 1/p .
By the self-improvement of reverse H¨older inequality (see [12, Appendix B]), we deduce that − Z B |h|p+εdµ 1/p . − Z 2B |h|2dµ 1/2 . In particular due to the definition of h, we have
− Z B |Γe−tLg|p+εdµ 1/p . − Z 2B |Γe−tLg|2dµ 1/2 + c(g, B),
which is the desired inequality.
5. Boundedness of Riesz transforms under elliptic perturbation Consider (M, d, µ) a doubling Riemannian manifold (satisfying (VDν)), equipped
with the Riemannian gradient ∇ and its divergence operator div = ∇∗, such that the self-adjoint Laplacian is given by ∆ := div∇ and for every f, g ∈ D(∆) we have
Z M f ∆(g) dµ = Z M h∇f (x), ∇g(x)iTxMdµ(x), where TxM is the tangent space of M at x.
Let A = A(x) be a complex matrix - valued function, defined on M and satisfying the ellipticity (or accretivity) condition
(5.1) λ|ξ|2 ≤ <hA(x)ξ, ξi and |hA(x)ξ, ζi| ≤ Λ|ξ||ζ|,
for some constants λ, Λ > 0 and every x ∈ M , ξ, ζ ∈ TxM .
To such a complex matrix valued function A, we may define a second order divergence form operator
which we first interpret in the sense of maximal accretive operators via a ses-quilinear form. That is, D(L) is the largest subspace contained in W1,2 := D(∇) for which Z M hA∇f, ∇gi dµ ≤ Ckgk2 ∀g ∈ W1,2, and we define Lf by hLf, gi = Z M hA∇f, ∇gi dµ for f ∈ D(L) and g ∈ W1,2. Thus defined, L = L
A is a maximal-accretive operator
on L2 and D(L) is dense in W1,2.
Let us motivate such considerations. Consider on the manifold (M, d, µ) two
complete Riemannian metrics Go = h, io and G = h, i. Assume that these two
metrics are quasi-isometric in the sense that the associated lengths | | and | |0 on
tangent vectors satisfy
c1|v|TxM ≤ |v|0,TxM ≤ c2|v|TxM
for some numerical constants c1, c2 and every x ∈ M and tangent vector v ∈ TxM .
This implies (see [11] and [20, Section 4]) that there exists an automorphism A = (A(x))x∈M of the tangent bundle T M such that for every x ∈ M and u, v ∈ TxM
hA(x)u, vi = hu, vi0.
Moreover, the matrix-valued map A satisfies the ellipticity condition (5.1). So there is an equivalence between the two points of view: to perturb the Laplace operator through the map A and to perturb the considered Riemannian metric on the manifold.
We denote by RHq(∆) and RHq(L) = RHq(LA) the reverse H¨older inequality
property for harmonic functions associated respectively with the Laplace operator ∆, or with the perturbed operator L = LA. For A and p ∈ (2, ∞), we set (Gp,A)
and (Rp,A) for the Lp-boundedness property of the gradient of the semigroup or the
Riesz transforms associated with the operator L = LA.
Let us also point out that in such a context it is well-known that (even if A is non self-adjoint) Lemma 2.10 holds.
Proposition 5.1. Let (M, d, µ) be a doubling Riemannian manifold with (1.1). Let us assume RHq(∆) for some q > 2. Then there exists ε0 > 0 such that for every
p ∈ (2, 2 + ε0) and every A with kA − Idk∞≤ ε0, Property RHp(LA) holds.
Proof. Consider a ball B0 and a function f ∈ D(LA), LA-harmonic on 2B0. Then
using Lemma 2.10 for every ball B ⊂ B0, there exists u ∈ W1,2 such that f − u ∈
W1,2is supported in the ball 2B and u is ∆-harmonic in 2B. Moreover, by repeating
the argument employed for Lemma 2.10, we have − Z B |∇(f − u)|2dµ = 1 |B| Z h∇(f − u), ∇(f − u)i dµ = 1 |B| Z M h∇(f − u), ∇f i dµ,
